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Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory

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  • Ogasawara, Haruhiko

Abstract

Asymptotic expansions of the distributions of the pivotal statistics involving log-likelihood derivatives under possible model misspecification are derived using the asymptotic cumulants up to the fourth-order and the higher-order asymptotic variance. The pivots dealt with are the studentized ones by the estimated expected information, the negative Hessian matrix, the sum of products of gradient vectors, and the so-called sandwich estimator. It is shown that the first three asymptotic cumulants are the same over the pivots under correct model specification with a general condition of the equalities. An application is given in item response theory, where the observed information is usually used rather than the estimated expected one.

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  • Ogasawara, Haruhiko, 2010. "Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2149-2167, October.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:9:p:2149-2167
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    1. R. Darrell Bock & Marcus Lieberman, 1970. "Fitting a response model forn dichotomously scored items," Psychometrika, Springer;The Psychometric Society, vol. 35(2), pages 179-197, June.
    2. Takesi Hayakawa, 1977. "The likelihood ratio criterion and the asymptotic expansion of its distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 29(1), pages 359-378, December.
    3. Haruhiko Ogasawara, 2003. "Asymptotic standard errors of irt observed-score equating methods," Psychometrika, Springer;The Psychometric Society, vol. 68(2), pages 193-211, June.
    4. Ogasawara, Haruhiko, 2000. "Asymptotic Standard Errors of IRT Equating Coefficients Using Moments," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 51(1), pages 1-23.
    5. Akahira Masafumi & Takeuchi Kei, 1982. "On Asymptotic Deficiency Of Estimators In Pooled Samples In The Presence Of Nuisance Parameters," Statistics & Risk Modeling, De Gruyter, vol. 1(1), pages 17-38, January.
    6. Haruhiko Ogasawara, 2009. "Asymptotic cumulants of the parameter estimators in item response theory," Computational Statistics, Springer, vol. 24(2), pages 313-331, May.
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    Cited by:

    1. Ke-Hai Yuan & Ying Cheng & Jeff Patton, 2014. "Information Matrices and Standard Errors for MLEs of Item Parameters in IRT," Psychometrika, Springer;The Psychometric Society, vol. 79(2), pages 232-254, April.
    2. Ogasawara, Haruhiko, 2013. "Asymptotic cumulants of ability estimators using fallible item parameters," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 144-162.
    3. Haruhiko Ogasawara, 2021. "A Unified Treatment of Agreement Coefficients and their Asymptotic Results: the Formula of the Weighted Mean of Weighted Ratios," Journal of Classification, Springer;The Classification Society, vol. 38(2), pages 390-422, July.
    4. Ogasawara, Haruhiko, 2015. "Asymptotic cumulants of some information criteria," ビジネス創造センターディスカッション・ペーパー (Discussion papers of the Center for Business Creation) 10252/5446, Otaru University of Commerce.
    5. Haruhiko Ogasawara, 2012. "Asymptotic expansions for the ability estimator in item response theory," Computational Statistics, Springer, vol. 27(4), pages 661-683, December.
    6. Ogasawara, Haruhiko, 2012. "Cornish-Fisher expansions using sample cumulants and monotonic transformations," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 1-18, January.
    7. Ogasawara, Haruhiko, 2013. "Asymptotic properties of the Bayes and pseudo Bayes estimators of ability in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 359-377.
    8. Ogasawara, Haruhiko, 2016. "Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 20-37.
    9. Ogasawara, Haruhiko, 2015. "Asymptotic cumulants of some information criteria (2nd version)," ビジネス創造センターディスカッション・ペーパー (Discussion papers of the Center for Business Creation) 10252/5497, Otaru University of Commerce.

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